What will happen to my husband for forging his signature on his father's checks?

Forgery is a crime of moral turpitude, and if convicted you can kiss your career aspirations goodbye. Hire a lawyer before you get duped into a plea you will regret for life.

This depends on whether the State is pressing charges. Also all the details of what happened.

Unfortunately, lawyers cannot see the future any better than you can. Your husband has probably committed felony forgery, but whether the prosecutor will exercise their discretion to bring lesser charges or no charges at all is up to the prosecutor. If the State is aware that a crime was committed, the victim does not need to "press charges" for a case to go forward. If charges have not yet issued, your husband should consider hiring a good lawyer to try and engage the prosecutor in pre-charging discussions to attempt to influence that charging decision.